Finite Group Databasebeta

← #29679594

Subgroups — #29679594

Order 15616.

1590 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D61C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3#21448198C211yesmaximal, characteristic
4C2×#16839619C211yesmaximal, characteristic
5#21445853C212yesmaximal, complemented
6#21445849C212yesmaximal, complemented
7#21445845C212yesmaximal, complemented
8#21445841C212yesmaximal, complemented
9#21445540C211yesmaximal, characteristic, complemented
10#21445538C211yesmaximal, characteristic, complemented
11C2×C61×#428C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×M16×D61—24no—
16C2×M16×D61—24no—
17C2×M16×D61C418yescomplemented
18C22×C8×D61C2212yes—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758198—21no—
22#16839804—24no—
23C2×#14519711C2212yescomplemented
24C2×#14519709C2212yescomplemented
25#16839623—21no—
26#16839619—21no—
27#16839619—21no—
28#16839619C2211yescharacteristic
29#16839619C2211yescharacteristic
30#16839619C2211yescharacteristic
31#16839619C2211yescharacteristic
32#16839619C2212yes—
33#16839619C2212yes—
34#16839617—21no—
35C4×#11758198—21no—
36C4×#11758198—21no—
37C2×#14519684—21no—
38#16839524C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41#16839524C2212yescomplemented
42C22×M16×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2211yescharacteristic
45C61×#428—24no—
46C2×#107×C61C2212yescomplemented
47C2×#106×C61C2212yescomplemented
48C24×D61D411yescharacteristic
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61C2×C414yes—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61Q814yes—
59C22×C4×D61C2311yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194C2×C414yes—
64C2×#11758189—24no—
65C2×#11758189—24no—
66C2×#11758189C2×C418yescomplemented
67M16×D61—232no—
68C22×#92418C2312yes—
69C2×C8×D61—24no—
70C2×C8×D61—24no—
71C2×C8×D61C2×C418yescomplemented
72#14519736—22no—
73#14519736—22no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—44no—
88C2×C4×Dic61—21no—
89#14519711—24no—
90#14519709—24no—
91#14519684—22no—
92#14519684—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61C2×C414yes—
100C2×M16×C61D414yes—
101C2×M16×C61D414yes—
102C22×C8×C61C2312yes—
103#117×C61—22no—
104#117×C61—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—21no—
113#107×C61—24no—
114#106×C61—24no—
115C24×C61C2×D411yescharacteristic
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—41no—
121C23×D61—41no—
122C23×D61C4⋊C414yes—
123C23×D61C2×D411yescharacteristic
124C23×D61C2×D411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×Q812yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C4⋊C414yes—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×C4×D61—28no—
151C2×C4×D61—28no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—28no—
157C2×C4×D61—28no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C4⋊C414yes—
167C2×C4×D61C4⋊C414yes—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×Q812yes—
171C2×C4×D61C2×Q812yes—
172M16×C61C22⋊C4116yescomplemented
173C2×C8×C61C22×C414yes—
174C2×C8×C61C2×D414yes—
175C2×C8×C61C2×D414yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C42×C61—42no—
185C22⋊Dic61—42no—
186#11758198—42no—
187#11758198—42no—
188#11758198—44no—
189#11758198—42no—
190#11758198—44no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4×Dic61—42no—
200#11758194—28no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418C22×C414yes—
204#11758189—216no—
205C8×D61—216no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61C2×C4⋊C412yes—
211C23×C61#12112yes—
212C23×C61#12112yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—28no—
225C22×D61—28no—
226C22×D61—44no—
227C22×D61—44no—
228C22×D61—44no—
229C22×D61—44no—
230C22×D61—44no—
231C22×D61C2×C4⋊C412yes—
232C22×D61C2×C4⋊C412yes—
233C22×D61AGL(1,ℤ/8)12yes—
234C22×D61#13712yes—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—84no—
245C2×C4×C61C2×C22⋊C414yes—
246C2×C4×C61C2×C4⋊C412yes—
247C2×C4×C61#11714yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61C22⋊Q814yes—
251C2×C4×C61C22⋊Q814yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—28no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—84no—
273C2×Dic61C2×C4⋊C412yes—
274C2×Dic61C2×C4⋊C412yes—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—28no—
284C4×D61—28no—
285C8×C61C2×C22⋊C418yescomplemented
286#92418—24no—
287#20661—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61#38014yes—
295C22×C61C22×C4⋊C411yescharacteristic
296C22×C61C2×#12111yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57511yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—416no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61#42814yescomplemented
317C4×C61—81no—
318C4×C61—82no—
319C4×C61—81no—
320C4×C61#38014yes—
321C4×C61C22×C22⋊C411yescharacteristic
322C4×C61C2×#11711yescharacteristic
323C4×C61C2×C22⋊Q811yescharacteristic
324C4×C61C2×C22⋊Q811yescharacteristic
325Dic61—22no—
326Dic61—22no—
327Dic61—81no—
328Dic61—82no—
329Dic61—81no—
330C23×M16—611no—
331C2×#428—611no—
332C2×#428D6111yescharacteristic, complemented
333#2280—611no—
334C2×#380—611no—
335#1809—612no—
336#1809—612no—
337#1808—612no—
338#1807—612no—
339#1736—611no—
340#1736—611no—
341C2×C61—24no—
342C2×C61C2×#38011yescharacteristic
343C2×C61C2×#42811yescharacteristic
344C2×C61#291111yescharacteristic
345C2×C61#305511yescharacteristic
346C2×C61#305511yescharacteristic
347C2×C61#305611yescharacteristic
348C2×C61#305611yescharacteristic
349D61—24no—
350D61—42no—
351C24×C4—611no—
352C22×M16—611no—
353C22×M16—618no—
354C22×M16—612no—
355C22×M16—1224no—
356C22×M16—1224no—
357C22×M16C2×D6111yescharacteristic
358C23×C8—612no—
359C2×#117—611no—
360C2×#117C2×D6111yescharacteristic
361C22×C4⋊C4—611no—
362C22×C4⋊C4C2×D6111yescharacteristic
363C22×C22⋊C4—1221no—
364#428—24no—
365#428—1224no—
366C2×#107—612no—
367C2×#107C2×D6112yescomplemented
368C2×#106—612no—
369C2×#106C2×D6112yescomplemented
370#381—1221no—
371#380—611no—
372#380—612no—
373#380—611no—
374#380—612no—
375#380—611no—
376#380—611no—
377#380—1221no—
378#380—1221no—
379#379—1221no—
380C4×C22⋊C4—1221no—
381C4×C22⋊C4—1221no—
382C2×#95—1221no—
383#340—612no—
384#340—612no—
385#340—612no—
386#340—612no—
387C61#2066111yescharacteristic, complemented, Sylow
388C25—611no—
389C23×C4—21no—
390C23×C4—611no—
391C23×C4—614no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—614no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—1222no—
398C23×C4—1222no—
399C23×C4—1222no—
400C23×C4—1222no—
401C23×C4—1221no—
402C23×C4C22×D6111yescharacteristic
403C2×M16—614no—
404C2×M16—618no—
405C2×M16—1222no—
406C2×M16—1222no—
407C2×M16—12232no—
408C2×M16—1224no—
409C2×M16—1224no—
410C2×M16C4×D6114yes—
411C2×M16D24414yes—
412C2×M16C22⋊D6114yes—
413C22×C8—612no—
414C22×C8—618no—
415C22×C8—1224no—
416C22×C8—1224no—
417C22×C8C22×D6112yes—
418#117—22no—
419#117—22no—
420#117—1222no—
421#117—1222no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6112yes—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1222no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—2444no—
443C2×C42—21no—
444C2×C42—1221no—
445#107—24no—
446#107—1224no—
447#106—24no—
448#106—1224no—
449#95—1222no—
450#95—1222no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455C24—611no—
456C24—611no—
457C24—614no—
458C24—1222no—
459C24—1222no—
460C24—1222no—
461C24—1222no—
462C24—2441no—
463C24—2441no—
464C24D4×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—614no—
479C22×C4—614no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1228no—
491C22×C4—1228no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1228no—
497C22×C4—1228no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4C2×C4×D6112yes—
505C22×C4C2×D24412yes—
506C22×C4D4×D6111yescharacteristic
507C22×C4#1175822412yes—
508C22×C4Q8×D6112yes—
509C22×C4#1175822612yes—
510C22×C4C2×C22⋊D6112yes—
511C22×C4C23×D6111yescharacteristic
512M16—1228no—
513M16—12216no—
514M16#11758198116yescomplemented
515C2×C8—614no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—12216no—
519C2×C8C2×C4×D6114yes—
520C2×C8C2×D24414yes—
521C2×C8C2×C22⋊D6114yes—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—1221no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4—1222no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—2442no—
536C22⋊C4—42no—
537C22⋊C4—2444no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C22⋊C4—2444no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—42no—
545C42—2442no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—612no—
551C23—612no—
552C23—612no—
553C23—612no—
554C23—1228no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1228no—
566C23—2444no—
567C23—2444no—
568C23—2444no—
569C23—2444no—
570C23—2444no—
571C23C4⋊C4×D6112yes—
572C23#1451974812yes—
573C23#1451975212yes—
574C23C2×D4×D6111yescharacteristic
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1228no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1228no—
601C2×C4—1228no—
602C2×C4—4884no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2442no—
612C2×C4C2×#1175819814yes—
613C2×C4C4⋊C4×D6112yes—
614C2×C4#1451973514yes—
615C2×C4#1451974812yes—
616C2×C4#1451975212yes—
617C2×C4#1451975314yes—
618C2×C4#1451975714yes—
619C2×C4C22×C4×D6111yescharacteristic
620C2×C4C22×C22⋊D6111yescharacteristic
621C2×C4C22×D24411yescharacteristic
622C2×C4C2×Q8×D6111yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822611yescharacteristic
625C8—1224no—
626C8C2×#1175819818yescomplemented
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—614no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22—24416no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22#1683961914yes—
649C22C2×C4⋊C4×D6111yescharacteristic
650C22C2×#1451974811yescharacteristic
651C22C2×#1451975211yescharacteristic
652C22AGL(1,ℤ/8)×D6111yescharacteristic
653C22#137×D6111yescharacteristic
654C22#1684055811yescharacteristic
655C22#1684056111yescharacteristic
656C4—81no—
657C4—82no—
658C4—81no—
659C4—1222no—
660C4—1222no—
661C4—4882no—
662C4—4881no—
663C4—4881no—
664C4#1683961914yes—
665C4C22×#1175819811yescharacteristic
666C4C2×#1451973511yescharacteristic
667C4C2×#1451975311yescharacteristic
668C4C2×#1451975711yescharacteristic
669C2—24no—
670C2—1224no—
671C2—2442no—
672C2C2×#1683961911yescharacteristic
673C2#428×D6111yescharacteristic
674C2#2145440611yescharacteristic
675C2#2145543211yescharacteristic
676C2#2145543911yescharacteristic
677C2#2145544311yescharacteristic
678C2#2145545011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#428D61
hypocenterC61#20661
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
5. #21445853C2semidirect
6. #21445849C2semidirect
7. #21445845C2semidirect
8. #21445841C2semidirect
9. #21445540C2semidirect
10. #21445538C2semidirect
11. C2×C61×#428C2semidirect
17. C2×M16×D61C4semidirect
23. C2×#14519711C22semidirect
24. C2×#14519709C22semidirect
38. #16839524C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
41. #16839524C22semidirect
46. C2×#107×C61C22semidirect
47. C2×#106×C61C22semidirect
66. C2×#11758189C2×C4semidirect
71. C2×C8×D61C2×C4semidirect
172. M16×C61C22⋊C4semidirect
285. C8×C61C2×C22⋊C4semidirect
316. C2×D61#428semidirect
332. C2×#428D61semidirect
367. C2×#107C2×D61semidirect
369. C2×#106C2×D61semidirect
387. C61#20661semidirect
514. M16#11758198semidirect
626. C8C2×#11758198semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×M16×D612
C2×#168398042
#214481982
C2×#168396192
#214458532
#214458492
#214458452
#214458412
#214455402
#214455382
C2×C61×#4282
#2066161